1. SRCOS Summer Research Conf, 2011 Multiple Testing Under Dependency, with Applications to Genomic Data Analysis Zhi Wei Department of Computer Science New Jersey Institute of Technology Joint work with Wenguang Sun

2. 2 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 Intro Microarray Measure the activity of thousands of genes simultaneously DE v.s. EE genes Notation: –X i ∈ {DE, EE} –Y i = (y i1, …, y im ; y i(m+1), …, y i(m+n) ), m, n replicates for 2 cond, resp. Phenotype 1 Phenotype 2 Expression X X X X X X Differential Expression (DE) Analysis Data for a single gene

3. 3 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 Microarray Time Course Data: A Motivating Example Microarray Time Course (MTC) data commonly collected for studying biological processes Tian, Nowak, and Brasier (2005) “A TNF-Induced Gene Expression Program Under Oscillatory NF-Kappab Control,” –study the biological process of how the cytokine tumor necrosis factor (TNF) initiates tissue inflammation Dataset: time-course microarray experiment to profile gene activities at 0hr, 1hr, 3hr, 6hr after inhibition of the NF-kB transcription factor To find: –(1) “Early response genes” that were differentially expressed (DE) less than 1hr after the NF-kB inhibition –(2) “Middle response genes” that were DE at 3hr but no response prior to that –(3) “Late response genes” that were DE at 6hr but no response prior to that –(4) “Biphasic genes” that were DE at both 1hr and 6hr hours, but not at 3hr.

4. 4 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 Existing solutions [Y i1,Y i2, …,Y iT ] XiXi X i1 X i2 …X iT Y i1 Y i2 …Y iT Solution 1: Find TDE genes –Moderated likelihood ratio statistic and Hotelling T 2 (Tai and Speed 06), –functional data analysis : Luan and Li (04), Storey et al. (05), Hong and Li (06) Solution 2: Find DE gene at each time point –T-test at each time point –HMM, Yuan and Kendziorski (06). Too detailed Too general ?

5. 5 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 Set-wise multiple testing Conceptualize the gene selection problem as a set-wise multiple testing problem –Test DE vs EE at each time point for each gene –Combine the tests across all time points as a set for DE patterns of interest, namely, one gene corresponding to a set of hypotheses Several issues in set-wise testing –(i) multiplicity: how to control testing errors (such as the false discovery rate) at the set level when thousands of gene are considered simultaneously; –(ii) optimality: how to optimally combine the testing results within a set so that sensitivity is maximized while multiplicity is controlled. –(iii) dependency: how to account for and exploit the dependency of the tests within a set, e.g. the high temporal correlation in MTC data.

6. 6 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 Related work Multiplicity issue for simultaneous set-wise tests –Formally addressed by (Benjamini & Heller, Biometrics, 2008) Test DE vs EE at each time point individually Combine the p-values at each time point into a pooled p-value by Simes BH procedure is applied to the Simes-combined P values –Partial conjunction tests Are all T hypotheses in the set true? (Conjunction null) Are at least u out of T hypotheses in the set false? (Partial Conjunction alternative) –Limitations Incapable of capturing sophisticated DE patterns, e.g., early response genes Not optimal because dependency is ignored, although FDR at set level (FSR) is controlled. Testing hypotheses under dependency (Sun & Cai, JRSSB, 2009) –One HMM to account for dependency of hypotheses –Local Index of Significance (LIS) statistics –Valid and optimal

7. 7 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 Characterizing Gene Temporal Expr. Patterns in a Set-wise Multiple Testing Framework Setup for MTC data: –Each gene has two possible states at each time point: equally expressed (EE) and differentially expressed (DE), denoted by 0 and 1, respectively. –Consider P sets of hypotheses: {(H i1,...,H iT ) : i = 1,...,P} for testing EE versus DE, where P is #genes and T, #time points. Pattern representation –2 T atomic patterns η ∈ {0,1} T, e.g. 00101, 11100, etc. for T=5 –Partition 2 T ηs into null Θ 0 and non-null Θ 1, the patterns of interest –2^(2 T ) -1 partitions  powerful temporal pattern characterization Conjunction: Θ 0 ={(η 1,..., η T ) ∈ {0, 1} T |∑ η t =0} Partial Conjunction: Θ 0 ={(η 1,..., η T ) ∈ {0, 1} T |∑ η t < u} “Late response genes”, DE after time t, Θ 1 ={(η 1,..., η T ) ∈ {0, 1} T | η 1 =…=η t =0 & η t+1 +…+η T >0}

8. 8 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 HMM The state sequence of gene i over time is a binary vector X i = (x i1,..., x iT ), where x it = 1 indicates that gene i at time t is DE and x it = 0 otherwise. Assume X i ∈ {0,1} T is distributed as a Markov chain –Transition probability can be either homogeneous or inhomogeneous Assume X i ┴ X j for i <>j Emission probability f(Y it |X it ) ~ Gamma-Gamma Model (Newton et al. 2001, Kendziorski et al 2003) X i1 X i2 …X iT Y i1 Y i2 …Y iT

9. 9 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 HMM-based set-wise testing Use EM algorithm for estimating all HMM parameters: initial prob., transition prob., emission prob. (Gamma-Gamma model) Define a binary vector ϑ = ( ϑ 1,..., ϑ p ) ∈ {0, 1} p, where ϑ i =1 if X i ∈ Patterns of Interest and ϑ i =0 otherwise. Define the generalized local index of significance (GLIS): X i1 X i2 …X iT Y i1 Y i2 …Y iT

10. 10 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 HMM-based set-wise testing (cont’d) GLIS testing procedure –Denote by GLIS (1),..., GLIS (p) the ordered GLIS values and H (1),..., H (p) the corresponding hypotheses –Let –Then reject all H (i), for i=1, 2,..., l –Valid and asymptotically optimal for set-wise multiple testing when the HMM parameter estimate is consistent (Sun and Wei, JASA, 2011) The GLIS statistic can be interpreted as the probability of a gene being null (i.e., not exhibiting a pattern of interest) given the observed expression data

11. 11 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 Simulation Setup –#subjects under both conditions are n 1 = n 2 = 10 –#genes P = 2000 –#time points =6 –#replications =200. –Nominal FSR level is 0.1. –HMM: π = (0.95, 0.05), A t = (a 00, 1−a 00 ; 1−a 11, a 11 ),  t = (α t,α 0t, ν t ). Dependency: fix  t = (10, 1, 0.5) and a 00 =0.95, change a 11 from 0.2 to.98 Gene expr. Signal: fix A t = (0.95, 0.05; 0.2, 0.8) and  t = (α t,1, 0.5), change α t Conjunction and Partial Conjunction Comparison –BH-Simes –Viterbi-based decision rule: the most probable state sequence η ∈ Θ 1 then select it. Evaluation: FSR (False Set Rate), MSR (Missed Set Rate) = 1 – sensitivity

12. 12 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 Conjunction Test Results

13. 13 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 Partial Conjunction Test Results

14. 14 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 Ranking efficiency A k = (0.95, 0.05; 0.2, 0.8) and  k = (α k,32, 4) Solid: Oracle GLIS Dashed: data-driven GLIS Dotted: BH-Simes

15. 15 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 Application Calvano et al (Nature 2005): A network-based analysis of systemic inflammation in humans. Goal: Identify genes in response to endotoxin Dataset: 4 healthy persons’ gene expression in whole blood leukocytes immediately before and at 2,4,6,9 and 24 h after the intravenous administration of bacterial endotoxin; 4 healthy persons w/o treatment as controls.

16. 16 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 Pattern Design Genes perturbed in response to treatments Sequentially perturbed genes Early and late response genes OR

17. 17 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 HMM Fitting

18. 18 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 Early Response Genes (DE from 2 nd time point)

19. 19 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 Late Response Genes (DE after 2 nd time point)

20. 20 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 Published Genome-Wide Associations through 3/2009, 398 published GWA at p < 5 x 10 -8 NHGRI GWA Catalog www.genome.gov/GWAStudies

21. 21 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 HMM-dependent (set-wise) hypothesis test in Genome-wide Association Studies (GWAS) Individual Tests (Wei, Sun et al Bioinformatics, 2009) –N chromosomes  N HMMs, emission prob: Normal mixtures –How to pool different HMMs together for having a global multiplicity control –Improvement: More reproducible, Higher sensitivity Set-wise Tests (Wang, Wei, and Sun, Statistics and Its Interface, 2010) –Testing a genomic region

22. 22 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 From HMM to Markov Random Field An undirected graph, where each node represents a random variable Xi and each edge represents a dependency Joint Distribution Function Markov Blanket and Conditional independence Inference –Exact inference is NP-complete –Approximation techniques: MCMC, loopy belief propagation, ICM

23. 23 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 Microarray DE analysis Discrete Markov Random Field Model for Latent Differential Expression States (Wei and Li, Bioinformatics, 2007) p genes on a network Gene expression Y i =(Y i1,…,Y im ; Y i(m+1),…,Y i(m+n) ) Phenotype 1 Phenotype 2 Networks built from KEGG: 1668 nodes (genes), 8011 edges Framework: f(X|Y) ~ f(Y|X)*Pr(X) Pr(X) ~ Markov random field (MRF) Model Emission Prob. f(Y|X) ~ Gamma-Gamma Model

24. 24 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 MRF in Genome-wide association studies (GWAS) GWAS data with network structure constructed based on LD dependency. (Li, Wei, and Maris, Biostatistics, 2010) Framework: f(X|Y) ~ f(Y|X)*Pr(X) X, disease assc. state: Pr(X) ~ Markov random field (MRF) Model Y, genotype counts, Emission Prob. f(Y|X) ~ Dirichlet-Trinomial Model

25. 25 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 MTC Data Hidden Spatial-Temporal MRF Model (Wei and Li, Annals of Applied Statistics 2008) Model both regulatory (gene network) and temporal dependency (time). g g -1 g+1 tt -1 t +1 Time Gene Y gt Gene Expression X gt = 0 or 1, Gene State Temporal Dependency Regulatory Dependency f(Y gt |X gt ) ~ Emission Prob. Spatial Neighbors Temporal Neighbors

26. 26 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 Challenges in MRF MRF-structured hypothesis test is valid and optimal? MRF-structured set-wise hypothesis test is valid optimal? Unlike HMM, the model fitting and parameter estimating for MRF is much more challenging –The EM algorithm for MRF is NP-hard –(Generalized) belief propagation algorithm

27. 27 New Jersey Inst. of Tech. SRCOS Summer Research Conf, 2011 References Wei Z and Li H (2007): A Markov random field model for network-based analysis of genomic data. Bioinformatics, 23: 1537-1544. Wei Z and Li H (2008): A spatial-temporal MRF model for network-based analysis of microarray time course gene expression data. Annals of Applied Statistics, 2: 408-429. Sun, W., and Cai, T. (2009). “Large-scale multiple testing under dependence”. Journal of the Royal Statistical Society, Series B, 71, 393-424. Wei Z, Sun W, Wang K and Hakonarson H (2009), Multiple Testing in Genome- Wide Association Studies via Hidden Markov Models, Bioinformatics, 25:2802- 2808 Li H, Wei Z and Maris J (2010), A Hidden Markov Random Field Model for Genome-wide Association Studies, Biostatistics, 11:139-150. Wang W, Wei Z, and Sun W (2010), Simultaneous Set-Wise Testing Under Dependence, with Applications to Genome-Wide Association Studies, Statistics and Its Interface, 3(4):501-512 Sun W and Wei Z (2011), Multiple Testing for Pattern Identification, with Applications to Microarray Time Course Experiments, Journal of the American Statistical Association, 106(493):73–88